Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{10r^3 - 30r^2 + 20r}{10r^2 + 90r - 100}$
First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {10r(r^2 - 3r + 2)} {10(r^2 + 9r - 10)} $ $ t = \dfrac{10r}{10} \cdot \dfrac{r^2 - 3r + 2}{r^2 + 9r - 10} $ Simplify: $ t = r \cdot \dfrac{r^2 - 3r + 2}{r^2 + 9r - 10}$ Next factor the numerator and denominator. $ t = r \cdot \dfrac{(r - 1)(r - 2)}{(r - 1)(r + 10)}$ Assuming $r \neq 1$ , we can cancel the $r - 1$ $ t = r \cdot \dfrac{r - 2}{r + 10}$ Therefore: $ t = \dfrac{ r(r - 2)}{ r + 10 }$, $r \neq 1$